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ϕ,iϕ,j\phi_{,i}\phi_{,j}

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What must therefore vanish, identically, is not the slope but the correctly defined curvature-frame component built from it. For any static metric of the form ds2s^2 = -U(x\mathbf x)^2\,dx02x_0^2 + δij\delta_{ij}\,dxix^i dxjx^j — flat spatial slices, all the position dependence carried by U — the only nonzero Christoffel symbols are Γ0i0\Gamma^0_{0i} = ∂i\partial_i ln⁡\ln U = ϕ,i\phi_{,i} and Γ00i\Gamma^i_{00} = U2U^2ϕ,i\phi_{,i} , both following directly from the metric’s definition. Substituting into the Riemann tensor’s defining combination for the mixed component RiR^i{}_{0j0} , every term built from purely spatial Christoffels vanishes because the spatial slices are exactly flat, and the two surviving pieces are…

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ϕ,i\phi_{,i}

Symbol phi_,i

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ϕ,j\phi_{,j}

Symbol phi_,j

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ϕ,iϕ,j\phi_{,i}\phi_{,j}

Equation 87 · Evolutionary Physics

The Bend an Elevator Cannot Fake

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

What must therefore vanish, identically, is not the slope but the correctly defined curvature-frame component built from it. For any static metric of the form ds2s^2 = -U(x\mathbf x)^2\,dx02x_0^2 + δij\delta_{ij}\,dxix^i dxjx^j — flat spatial slices, all the position dependence carried by U — the only nonzero Christoffel symbols are Γ0i0\Gamma^0_{0i} = ∂i\partial_i ln⁡\ln U = ϕ,i\phi_{,i} and Γ00i\Gamma^i_{00} = U2U^2ϕ,i\phi_{,i} , both following directly from the metric’s definition. Substituting into the Riemann tensor’s defining combination for the mixed component RiR^i{}_{0j0} , every term built from purely spatial Christoffels vanishes because the spatial slices are exactly flat, and the two surviving pieces are…

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ϕ,iϕ,j\phi_{,i}\phi_{,j}

Equation 111 · Evolutionary Physics

The Bend an Elevator Cannot Fake

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Two known limits anchor it. First, the elevator: Tij\mathcal T_{ij} ≡\equiv 0 for any uniformly accelerated frame, proven directly above — the “uniform potential difference is not curvature” claim, derived rather than asserted. Second, weak-field gravity: expanding ϕ\phi ≈\approx Φ\Phi/c2c^2 for small Φ\Phi/c2c^2 , the quadratic term ϕ,i\phi_{,i}ϕ,j\phi_{,j} is second order in the already-small ratio Φ\Phi/c2c^2 and negligible next to the first-order term ϕ,ij\phi_{,ij} ≈\approx Φ,ij\Phi_{,ij}/c2c^2 , so Tij\mathcal T_{ij} →\to Φ,ij\Phi_{,ij} , exactly the ordinary Newtonian tidal tensor of tower and satellite geodesy [ 15 , 14 ] . The first derivative alone, meanwhile, recovers the textbook gravitational redshift these networks…

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